Helix of a curve
Let be a parametrized curve on a manifold . The helix or spiral of is a parametrized curve in that sends in the parameter domain to . In other words, the helix of a curve is the Cartesian product of the curve with the identity-parametrized curve.
Note that the above definition requires a parametrized curve. However, for a Riemannian manifold, any smooth curve in the manifold acquires a natural parametrization by arc-length, and we can thus give a natural choice of helix, or spiral, associated with the curve.
For a closed curve, viz a map , we construct the helix using the map that composes the modulo map with the curve parametrization.